To find the direction of the vector sum $\vec{A} + \vec{B}$, we can break down each vector into its horizontal ($x$) and vertical ($y$) components, sum these components, and then use trigonometry to find the resultant vector's direction.

Step-by-Step Solution

1. Convert each vector to components:

   • Vector $\vec{A}$:

     • Magnitude $A = 31.1 \, \text{m}$
     • Angle $\theta_A = 32.3^\circ$
     • $A_x = A \cos \theta_A = 31.1 \cos(32.3^\circ)$
     • $A_y = A \sin \theta_A = 31.1 \sin(32.3^\circ)$

   • Vector $\vec{B}$:

     • Magnitude $B = 28.7 \, \text{m}$
     • Angle $\theta_B = 142^\circ$
     • $B_x = B \cos \theta_B = 28.7 \cos(142^\circ)$
     • $B_y = B \sin \theta_B = 28.7 \sin(142^\circ)$

2. Calculate the components:

   • For $\vec{A}$: $A_x = 31.1 \cos(32.3^\circ)$ $A_y = 31.1 \sin(32.3^\circ)$

   • For $\vec{B}$: $B_x = 28.7 \cos(142^\circ)$ $B_y = 28.7 \sin(142^\circ)$

3. Add the components:

   • $R_x = A_x + B_x$
   • $R_y = A_y + B_y$

4. Find the direction of the resultant vector $\vec{R} = \vec{A} + \vec{B}$:

   • The direction (angle) $\theta_R$ of $\vec{R}$ is given by: $\theta_R = \tan^{-1} \left( \frac{R_y}{R_x} \right)$

Let’s calculate these values explicitly.Here are the computed components and the resultant vector's direction:

1. Components of $\vec{A}$:

   • $A_x = 26.29 \, \text{m}$
   • $A_y = 16.62 \, \text{m}$

2. Components of $\vec{B}$:

   • $B_x = -22.62 \, \text{m}$
   • $B_y = 17.67 \, \text{m}$

3. Resultant vector components $\vec{R} = \vec{A} + \vec{B}$:

   • $R_x = 3.67 \, \text{m}$
   • $R_y = 34.29 \, \text{m}$

4. Direction of $\vec{R}$:

   • $\theta_R = 83.89^\circ$

Conclusion

The direction of the vector sum $\vec{A} + \vec{B}$ is approximately 83.89 degrees.

Would you like more details on any specific step, or have any questions?

Here are some additional related questions:

1. How do you calculate the magnitude of a resultant vector?
2. What is the significance of the direction angle in vector addition?
3. How would the result change if the vectors were subtracted instead?
4. How can we use vector components to solve for resultant vectors in three dimensions?
5. Why is it important to convert angles to radians in trigonometric calculations?

Tip: When dealing with vector addition, always ensure you’re consistent with angle measurements (degrees or radians) throughout the calculations.